Is there a way to rotate the origin without rotating the geometry?
First go into Edit Mode (Tab). Select the whole mesh (A) and copy it to the buffer (CtrlC).



Leave Edit Mode Tab and rotate (R) your object as you like. Then go back into Edit Mode, delete the whole mesh (X or DEL).



While still in Edit Mode, paste the mesh from the buffer (Space and type "Paste Selection from Buffer" then hit Enter).



Now leave Edit Mode and the mesh is back in place, but the object is rotated.



Not a direct button to do what you want, but this is pretty quick to do.